A weighted Plancherel formula II. The case of the ball

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Abstract

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The group SU(1,d) acts naturally on the Hilbert space L²(Bdμα)(α>-1), where B is the unit ball of ℂd and dμα the weighted measure (1-|z|²)αdm(z). It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic tensor fields.